Problem: Simplify and expand the following expression: $ \dfrac{2}{k + 9}+ \dfrac{4}{k - 4}- \dfrac{5}{k^2 + 5k - 36} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{5}{k^2 + 5k - 36} = \dfrac{5}{(k + 9)(k - 4)}$ Now we have: $ \dfrac{2}{k + 9}+ \dfrac{4}{k - 4}- \dfrac{5}{(k + 9)(k - 4)} $ The least common multiple of the denominators is: $ (k + 9)(k - 4)$ In order to get the first term over $(k + 9)(k - 4)$ , multiply by $\dfrac{k - 4}{k - 4}$ $ \dfrac{2}{k + 9} \times \dfrac{k - 4}{k - 4} = \dfrac{2(k - 4)}{(k + 9)(k - 4)} $ In order to get the second term over $(k + 9)(k - 4)$ , multiply by $\dfrac{k + 9}{k + 9}$ $ \dfrac{4}{k - 4} \times \dfrac{k + 9}{k + 9} = \dfrac{4(k + 9)}{(k + 9)(k - 4)} $ Now we have: $ \dfrac{2(k - 4)}{(k + 9)(k - 4)} + \dfrac{4(k + 9)}{(k + 9)(k - 4)} - \dfrac{5}{(k + 9)(k - 4)} $ $ = \dfrac{ 2(k - 4) + 4(k + 9) - 5} {(k + 9)(k - 4)} $ Expand: $ = \dfrac{2k - 8 + 4k + 36 - 5}{k^2 + 5k - 36} $ $ = \dfrac{6k + 23}{k^2 + 5k - 36}$